A Stability Algorithm for a Special Case of the Milling Process: Contribution to Machine Tool Chatter Research—6

1968 ◽  
Vol 90 (2) ◽  
pp. 325-329 ◽  
Author(s):  
R. E. Hohn ◽  
R. Sridhar ◽  
G. W. Long
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Machine Tool ◽  
Linear Differential Equation ◽  
Mathieu Equation ◽  
Milling Process ◽  
Machine Tool Chatter ◽  
Linear Differential ◽  
The Stability ◽  
Special Case

In an effort to determine the stability of the milling process, and due to the complexity of its describing equation, a special case of this equation is considered. In this way, it is possible to isolate and study its salient characteristics. Moreover, the simplified equation is representative of a machining operation on which experimental data can be obtained. This special case is described by a linear differential equation with periodic coefficients. A computer algorithm is developed for determining the stability of this equation. To demonstrate the use of the algorithm on an example whose solution is known, the classical Mathieu equation is studied. Also, experimental results on an actual machining operation described by this type of equation are compared to the results found using the stability algorithm. As a result of this work, some knowledge about the stability solution of the general milling process is obtained.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

scholarly journals Methods for stability and instability of the solution of one non-linear differential equation

1996 ◽  
Vol 18 (3) ◽  
pp. 1-7
Author(s):  
Nguyen Dang Bich ◽  
Nguyen Vo Thong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Analytical Solutions ◽  
Flexible Structures ◽  
Stability Conditions ◽  
Stability And Instability ◽  
Non Linear ◽  
Linear Differential ◽  
The Stability

When research on instability states of tall and flexible structures is carried out, the solutions of non-linear differential equations have to be investigated. Although the fully analytical solutions of the moving rule of the structures cannot be found, but based on the conditions applied to the parameters of the moving differential equations the authors have studied the characteristics of the solutions when t→∞. Then the instability of the structures may be investigated, and the stability conditions can be concluded. This is the content which this paper would like to present. 

scholarly journals Whit ham method of the expansion near the wave front propagating on the river with bed load transport

1990 ◽  
Vol 12 (3) ◽  
pp. 8-14
Author(s):  
Pham Hung
Keyword(s):  
Differential Equation ◽  
Wave Front ◽  
Water Surface ◽  
Bed Load ◽  
Flood Wave ◽  
Load Transport ◽  
Bed Load Transport ◽  
Linear Differential ◽  
The Stability ◽  
Special Case

In this paper the non linear differential equation describing the behavior of water surface behind the wave front is obtained by Whit ham method. It is shown that the bed load transport could influence significantly on the stability of the flood wave. In the supercritical flow, the bed load wave will propagate upstream and will linearly stabilize. The different situations when the wave could topple over are analyzed. In the flow without the bed load transport, the results of Whit ham have been reobtained as a special case.  

A Stability Algorithm for the General Milling Process: Contribution to Machine Tool Chatter Research—7

1968 ◽  
Vol 90 (2) ◽  
pp. 330-334 ◽  
Author(s):  
R. Sridhar ◽  
R. E. Hohn ◽  
G. W. Long
Keyword(s):  
Stability Analysis ◽  
Milling Process ◽  
Stability Boundaries ◽  
Machine Tool Chatter ◽  
Milling Operation ◽  
Cutting Operation ◽  
Differential Difference Equation ◽  
Linear Differential ◽  
The Stability

In this paper, a method of stability analysis for the general milling process is given. The milling operation is described by a linear differential-difference equation with periodic coefficients. An algorithm which can be used in conjunction with the digital computer is developed as a means of analytically determining the stability of this equation. This algorithm will permit the determination of the stability boundaries in the space of controllable parameters associated with a cutting operation and allows more realistic models for milling to be studied than have been attempted up to the present time. The technique is used to predict the stability in an example of a milling operation.

A system of order 2 in which small sub-harmonics and large sub-harmonics coexist

1988 ◽  
Vol 108 (1-2) ◽  
pp. 35-44
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
General Theory ◽  
Linear Differential Equation ◽  
Special Form ◽  
Small Parameters ◽  
Linear Differential ◽  
Image Position ◽  
Small Periodic ◽  
Special Case

SynopsisThis paper establishes the coexistence of small and large subharmonics in a special case of the ordinary non-linear differential equationof order 2, where κ, ε are small parameters, λ>0 is a parameter independent of κ, ε, h(t) has the least period 2π andIt is divided into three sections. In Section 1 a general analysis of the periodic solutions of (.), classified into small, medium or large, is given. In Section 2 the general theory of Section 1 is applied to the special form of (.) where k = vε1+s, v>0, s>0 constants, to obtain results from which we extract in Section 2 a theorem (Section 3) on the coexistence of a small periodic solution of order 1, several small sub-harmonics and several large sub-harmonics of the special caseof (.), where Q≧1 is an integer.

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